Question Number 168732 by mnjuly1970 last updated on 16/Apr/22
Answered by mnjuly1970 last updated on 16/Apr/22
$$\:\:\:\:\:\:\:\:\Omega\:=\:\int_{\mathrm{0}} ^{\:\infty} \frac{\:{x}^{\:\frac{\mathrm{1}}{\mathrm{4}}} +\:\mathrm{2}{x}^{\:\:\frac{\mathrm{5}}{\mathrm{4}}} +{x}^{\:\frac{\mathrm{9}}{\mathrm{4}}} }{\left(\mathrm{1}+{x}^{\:\mathrm{3}} \right)^{\:\mathrm{2}} }{dx} \\ $$$$\:\:\:\:\:\:\:\:\:\overset{{x}^{\:\mathrm{3}} =\:{t}} {=}\:\frac{\mathrm{1}}{\mathrm{3}}\int_{\mathrm{0}} ^{\:\infty} \frac{{t}^{\:\frac{\mathrm{1}}{\mathrm{12}}−\frac{\mathrm{8}}{\mathrm{12}}} \:+\:\mathrm{2}{t}^{\:\frac{\mathrm{5}}{\mathrm{12}}−\frac{\mathrm{8}}{\mathrm{12}}} +\:{t}^{\:\frac{\mathrm{9}}{\mathrm{12}}−\frac{\mathrm{8}}{\mathrm{12}}} }{\left(\mathrm{1}\:+\:{t}\:\right)^{\:\mathrm{2}} }\:{dt} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:=\:\frac{\mathrm{1}}{\mathrm{3}}\:\int_{\mathrm{0}} ^{\:\infty} \frac{{t}^{\:\frac{−\mathrm{7}}{\mathrm{12}}} +\mathrm{2}{t}^{\:\frac{−\mathrm{1}}{\mathrm{4}}} +\:{t}^{\:\frac{−\mathrm{1}}{\mathrm{12}}} \:}{\left(\mathrm{1}+\:{t}\:\right)^{\:\mathrm{2}} }{dt} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:=\:\frac{\mathrm{1}}{\mathrm{3}\:}\:\int_{\mathrm{0}} ^{\:\infty} \:\frac{{t}^{\:\frac{\mathrm{5}}{\mathrm{12}}\:−\mathrm{1}} +\:\mathrm{2}{t}^{\:\frac{\mathrm{3}}{\mathrm{4}}\:−\mathrm{1}} +\:{t}^{\:\frac{\mathrm{11}}{\mathrm{12}}−\mathrm{1}} }{\left(\mathrm{1}\:+\:{t}\:\right)^{\:\mathrm{2}} }{dt} \\ $$$$\:\:\:\:\:\:\:\:\:\:=\:\frac{\mathrm{1}}{\mathrm{3}}\:\left\{\:\beta\:\left(\frac{\mathrm{5}}{\mathrm{12}}\:,\:\frac{\mathrm{19}}{\mathrm{12}}\right)\:+\mathrm{2}\beta\left(\frac{\mathrm{3}}{\mathrm{4}}\:,\:\frac{\mathrm{5}}{\mathrm{4}}\right)\:+\beta\:\left(\frac{\mathrm{11}}{\mathrm{12}}\:,\:\frac{\mathrm{13}}{\mathrm{12}}\:\right\}\right. \\ $$$$\:\:\:\:\:\:=\:\frac{\mathrm{1}}{\mathrm{3}}\:\left\{\:\frac{\Gamma\left(\frac{\mathrm{5}}{\mathrm{12}}\:\right)\Gamma\left(\:\frac{\mathrm{19}}{\mathrm{12}}\right)+\mathrm{2}\Gamma\left(\frac{\mathrm{3}}{\mathrm{4}}\right)\Gamma\left(\frac{\mathrm{5}}{\mathrm{4}}\right)+\Gamma\:\left(\frac{\mathrm{11}}{\mathrm{12}}\right)\Gamma\:\left(\frac{\mathrm{13}}{\mathrm{12}}\:\right)\:}{\left[\Gamma\left(\mathrm{2}\right)=\:\mathrm{1}\:\right]}\right\} \\ $$$$\:\:\:\:\:\:=\:\frac{\mathrm{1}}{\mathrm{3}}\:\left\{\:\frac{\mathrm{7}}{\mathrm{12}}\:\frac{\pi}{{sin}\left(\frac{\mathrm{5}\pi}{\mathrm{12}}\right)}\:+\frac{\mathrm{1}}{\mathrm{2}}\:\frac{\pi}{{sin}\left(\frac{\pi}{\mathrm{4}}\right)}\:+\frac{\mathrm{1}}{\mathrm{12}}\:\frac{\pi}{{sin}\left(\frac{\pi}{\mathrm{12}}\right)}\:\right\} \\ $$$$\:\:\:\:\:\:=\:\frac{\mathrm{1}}{\mathrm{3}}\:\left\{\:\frac{\mathrm{7}}{\mathrm{3}\:}\:\frac{\pi}{\:\sqrt{\mathrm{6}}\:+\sqrt{\mathrm{2}}}\:+\:\frac{\pi}{\:\sqrt{\mathrm{2}}}\:+\frac{\mathrm{1}}{\mathrm{3}}\:\frac{\pi}{\:\sqrt{\mathrm{6}}\:−\sqrt{\mathrm{2}}}\:\right\} \\ $$$$\:\:\:\:\:\:=\:\frac{\pi}{\mathrm{3}}\:\left\{\:\frac{\mathrm{7}\left(\sqrt{\mathrm{6}}\:−\sqrt{\mathrm{2}}\:\right)}{\mathrm{12}}\:+\frac{\mathrm{1}}{\:\sqrt{\mathrm{2}}}\:+\:\frac{\sqrt{\mathrm{6}}\:+\sqrt{\mathrm{2}}}{\mathrm{12}}\:\right\} \\ $$$$\:\:\:\:\:\:=\:\frac{\pi}{\mathrm{3}}\:\left\{\:\frac{\mathrm{8}\sqrt{\mathrm{6}}\:−\mathrm{6}\sqrt{\mathrm{2}}\:}{\mathrm{12}}\:+\:\frac{\mathrm{1}}{\:\sqrt{\mathrm{2}}\:}\:\right\} \\ $$$$\:\:\:\:\:\:=\:\frac{\pi}{\mathrm{3}}\left(\frac{\mathrm{8}\sqrt{\mathrm{6}}}{\mathrm{12}}\right)=\:\frac{\mathrm{2}}{\mathrm{3}}\:\frac{\sqrt{\mathrm{6}}}{\mathrm{3}}\pi\:=\:\frac{\mathrm{2}}{\mathrm{3}}\sqrt{\frac{\mathrm{2}}{\mathrm{3}}}\:\pi \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\therefore\:\Omega\:=\:\sqrt{\frac{\mathrm{4}}{\mathrm{9}}}\:\sqrt[{\mathrm{4}}]{\left(\frac{\mathrm{4}}{\mathrm{9}}\right)}\:\pi \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{a}\:−\:\mathrm{1}\:=\:\frac{\mathrm{4}}{\mathrm{9}}\:\Rightarrow\:{a}\:=\:\frac{\mathrm{13}}{\mathrm{9}}\:\:\:\:\:\:\:\checkmark \\ $$$$\:\: \\ $$
Commented by safojontoshtemirov last updated on 16/Apr/22
very nice
Commented by Tawa11 last updated on 16/Apr/22
$$\mathrm{Great}\:\mathrm{sir}. \\ $$
Answered by Mathspace last updated on 17/Apr/22
$$\Psi=\int_{\mathrm{0}} ^{\infty} \:\frac{{x}^{−\frac{\mathrm{1}}{\mathrm{4}}} }{\left({x}^{\mathrm{2}} −{x}+\mathrm{1}\right)^{\mathrm{2}} }{dx}\:\Rightarrow \\ $$$$\Psi=\int_{\mathrm{0}} ^{\infty} \:\:\frac{{x}^{−\frac{\mathrm{1}}{\mathrm{4}}} }{\left(\frac{{x}^{\mathrm{3}} +\mathrm{1}}{{x}+\mathrm{1}}\right)^{\mathrm{2}} }{dx} \\ $$$$=\int_{\mathrm{0}} ^{\infty} \:\frac{{x}^{−\frac{\mathrm{1}}{\mathrm{4}}} \left({x}^{\mathrm{2}} +\mathrm{2}{x}+\mathrm{1}\right)}{\left(\mathrm{1}+{x}^{\mathrm{3}} \right)^{\mathrm{2}} }{dx} \\ $$$$=\int_{\mathrm{0}} ^{\infty} \:\:\frac{{x}^{\frac{\mathrm{7}}{\mathrm{4}}} +\mathrm{2}{x}^{\frac{\mathrm{3}}{\mathrm{4}}} \:+{x}^{−\frac{\mathrm{1}}{\mathrm{4}}} }{\left(\mathrm{1}+{x}^{\mathrm{3}} \right)^{\mathrm{2}} }{dx} \\ $$$${let}\:\varphi\left({a},{b}\right)=\int_{\mathrm{0}} ^{\infty} \:\:\frac{{x}^{{a}−\mathrm{1}} }{{x}^{\mathrm{3}} +{b}^{\mathrm{3}} }{dx}\:\Rightarrow \\ $$$$\frac{\partial\varphi}{\partial{b}}=−\mathrm{3}{b}^{\mathrm{2}} \int_{\mathrm{0}} ^{\infty} \:\frac{{x}^{{a}−\mathrm{1}} }{\left({x}^{\mathrm{3}} +{b}^{\mathrm{3}} \right)^{\mathrm{2}} }{dx}\:\Rightarrow \\ $$$$\int_{\mathrm{0}} ^{\infty} \:\:\frac{{x}^{{a}−\mathrm{1}} }{\left({x}^{\mathrm{3}} +\mathrm{1}\right)^{\mathrm{2}} }{dx}=−\frac{\mathrm{1}}{\mathrm{3}}\frac{\partial\varphi}{\partial{b}}\mid_{{b}=\mathrm{1}} \\ $$$$\varphi\left({a},{b}\right)=_{{x}={bz}} \:\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{\left({bz}\right)^{{a}−\mathrm{1}} }{{b}^{\mathrm{3}} \left(\mathrm{1}+{z}^{\mathrm{3}} \right)}{b}\:{dz} \\ $$$$={b}^{{a}−\mathrm{3}} \int_{\mathrm{0}} ^{\infty} \:\:\frac{{z}^{{a}−\mathrm{1}} }{\mathrm{1}+{z}^{\mathrm{3}} }{dz}\:\:\:\:\:\left({z}^{\mathrm{3}} ={t}\right) \\ $$$$={b}^{{a}−\mathrm{3}} \int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{t}^{\frac{{a}−\mathrm{1}}{\mathrm{3}}} }{\mathrm{1}+{t}}\frac{\mathrm{1}}{\mathrm{3}}{t}^{\frac{\mathrm{1}}{\mathrm{3}}−\mathrm{1}} \:{dt} \\ $$$$=\frac{{b}^{{a}−\mathrm{3}} }{\mathrm{3}}\int_{\mathrm{0}} ^{\infty} \:\:\frac{{t}^{\frac{{a}}{\mathrm{3}}−\mathrm{1}} }{\mathrm{1}+{t}}{dt} \\ $$$$=\frac{{b}^{{a}−\mathrm{3}} }{\mathrm{3}}×\frac{\pi}{{sin}\left(\frac{{a}\pi}{\mathrm{3}}\right)}\:\Rightarrow \\ $$$$\frac{\partial\varphi}{\partial{b}}\left({a},{b}\right)=\frac{\left({a}−\mathrm{3}\right.}{\mathrm{3}}{b}^{{a}−\mathrm{4}} \frac{\pi}{{sin}\left(\frac{{a}\pi}{\mathrm{3}}\right)}\:\Rightarrow \\ $$$$\frac{\partial\varphi}{\partial{b}}\mid_{{b}=\mathrm{1}} =\frac{{a}−\mathrm{3}}{\mathrm{3}}×\frac{\pi}{{sin}\left(\frac{\pi{a}}{\mathrm{3}}\right)} \\ $$$$\Rightarrow\int_{\mathrm{0}} ^{\infty} \frac{{x}^{{a}−\mathrm{1}} }{\left({x}^{\mathrm{3}} +\mathrm{1}\right)^{\mathrm{2}} }{dx}=\frac{\pi\left(\mathrm{3}−{a}\right)}{\mathrm{9}{sin}\left(\frac{\pi{a}}{\mathrm{3}}\right)} \\ $$$$\Rightarrow\int_{\mathrm{0}} ^{\infty} \frac{{x}^{\frac{\mathrm{7}}{\mathrm{4}}} }{\left(\mathrm{1}+{x}^{\mathrm{3}} \right)^{\mathrm{2}} }{dx}\:\:\:=\int_{\mathrm{0}} ^{\infty} \:\frac{{x}^{\frac{\mathrm{11}}{\mathrm{4}}−\mathrm{1}} }{\left(\mathrm{1}+{x}^{\mathrm{3}} \right)^{\mathrm{2}} }{dx} \\ $$$$=\frac{\pi\left(\mathrm{3}−\frac{\mathrm{11}}{\mathrm{4}}\right)}{\mathrm{9}{sin}\left(\frac{\mathrm{11}\pi}{\mathrm{12}}\right)}=… \\ $$$$\int_{\mathrm{0}} ^{\infty} \:\:\frac{{x}^{\frac{\mathrm{3}}{\mathrm{4}}} }{\left(\mathrm{1}+{x}^{\mathrm{3}} \right)^{\mathrm{2}} }{dx}=\int_{\mathrm{0}} ^{\infty} \:\frac{{x}^{\frac{\mathrm{7}}{\mathrm{4}}−\mathrm{1}} }{\left(\mathrm{1}+{x}^{\mathrm{3}} \right)^{\mathrm{2}} }{dx} \\ $$$$=\frac{\pi\left(\mathrm{3}−\frac{\mathrm{7}}{\mathrm{4}}\right)}{\mathrm{9}{sin}\left(\frac{\mathrm{7}\pi}{\mathrm{12}}\right)}=… \\ $$